A forgotten paper on the fundamental theorem of algebra

2000 ◽  
Vol 54 (3) ◽  
pp. 333-341 ◽  
Author(s):  
F. Smithies
Keyword(s):  
Fundamental Theorem ◽  
Royal Society ◽  
Complex Form ◽  
Careful Examination ◽  
Complex Field ◽  
Complex Zero ◽  
Original Idea ◽  
Fundamental Theorem Of Algebra ◽  
Complex Coefficients

In 1798, there appeared in the Philosophical Transactions of the Royal Society a paper by James Wood, purporting to prove the fundamental theorem of algebra, to the effect that every non-constant polynomial with real coefficients has at least one real or complex zero. Since the first generally accepted proof of this result was given by Gauss in 1799, Wood's paper deserves careful examination. After giving a brief outline of Wood's career, I describe the argument of his paper. His proof turns out to be incomplete as it stands, but it contains an original idea, which was to be used later, in the same context, by von Staudt, Gordan and others, without knowledge of Wood's work. After putting Wood's work in context, I conclude by showing how his idea can be used to prove the complex form of the fundamental theorem of algebra, stating that every non-constant polynomial with complex coefficients has at least one zero in the complex field.

Activities for Students: A Graphical Approach to Understanding the Fundamental Theorem of Algebra

2001 ◽  
Vol 94 (9) ◽  
pp. 749-756
Author(s):  
Sudhir Kumar Goel ◽  
Denise T. Reid
Keyword(s):  
Fundamental Theorem ◽  
Polynomial Equation ◽  
Complex Numbers ◽  
Multiple Roots ◽  
Complex Root ◽  
Graphical Approach ◽  
Fundamental Theorem Of Algebra ◽  
Complex Coefficients

The fundamental theorem of algebra states, Every polynomial equation of degree n ≥ 1 with complex coefficients has at least one complex root. This fact implies that these equations have exactly n roots, counting multiple roots, in the set of complex numbers.

Delving Deeper: Decomposing Quintics

2009 ◽  
Vol 102 (9) ◽  
pp. 710-713
Author(s):  
Raghavendra G. Kulkarni
Keyword(s):  
Fundamental Theorem ◽  
Rational Numbers ◽  
Lower Degree ◽  
Polynomial Factorization ◽  
Complex Field ◽  
Existence Proof ◽  
Integer Coefficients ◽  
Polynomial Decomposition ◽  
Fundamental Theorem Of Algebra ◽  
Completely Reducible

Polynomial decomposition, also referred to as polynomial factorization, is the process of splitting a given polynomial of degree n into its constituent factors—that is, polynomials of lower degree. A reducible polynomial over a given field—such as the real (ℝ), complex (ℂ), or rational numbers (ℚ)—is one that can be factored into polynomials of lower degree with coefficients in that field; otherwise, it is irreducible over the field (Thangadurai 2007). From the fundamental theorem of algebra, we know that a polynomial with integer coefficients is completely reducible into linear factors over the complex field, whereas it is reducible to linear and quadratic factors over ℝ; however, it may be irreducible over ℚ. If a polynomial (with rational coefficients) can be factored over ℚ, Gauss's lemma states, it can be factored over the integers as well. The fundamental theorem of algebra is only an existence proof and does not provide any procedure for factoring a polynomial.

scholarly journals XIV. On the absorption of gases by solids

1881 ◽  
Vol 32 (212-215) ◽  
pp. 407-408
Keyword(s):  
Royal Society ◽  
Hydraulic Press ◽  
Careful Examination ◽  
Solid Matter ◽  
Chemical Action ◽  
High Pressure And Temperature ◽  
Solid Iron ◽  
Series Of Experiments ◽  
The Subject ◽  
Very High

During the progress of the investigations which I have from time to time had the honour of bringing under the notice of the Royal Society, I have again and again noticed the apparent disappearance of gases inclosed in vessels of various materials when the disappearance could not be accounted for upon the assumption of ordinary leakage. After a careful examination of the subject I found that the solids absorbed or dissolved the gases, giving rise to a striking example of the fixation of a gas in a solid without chemical action. In carrying out that most troublesome investigation, the crystalline separation of carbon from its compounds, the tubes used for experiment have been in nine cases out of ten found to be empty on opening them, and in most cases a careful testing by hydraulic press showed no leakage. The gases seemed to go through the solid iron, although it was 2 inches thick. A series of experiments with various linings were tried. The tube was electro-plated with copper, silver, and gold, but with no greater success. Siliceous linings were tried fusible enamels and glass—but still the' tubes refused to hold the contents. Out of thirty-four experiments made since my last results were published, only four contained any liquid or condensed gaseous matter after the furnacing. I became convinced that the solid matter at the very high pressure and temperature used must be pervious to gases.

Sign in / Sign up

Export Citation Format

Share Document